#6583963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,26,34)(16,20,29,37)(17,21,30,38)(23,31,25,33)(24,32,40,43)(27,35,28,36)(39,42,41,44);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,32)(21,31)(22,34)(25,30)(27,39)(28,41)(29,40)(33,38)(35,42)(36,44)(37,43);(1,2,3,4)(15,23)(16,27)(17,30)(19,31)(20,35)(21,38)(25,26)(28,29)(33,34)(36,37)(39,41)(42,44);(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,24)(17,27)(18,29)(19,32)(21,35)(22,37)(23,39)(25,41)(26,40)(28,30)(31,42)(33,44)(34,43)(36,38);(15,25)(16,27)(17,30)(19,33)(20,35)(21,38)(23,26)(24,40)(28,29)(31,34)(32,43)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491159);